# Thread: Help with solving three trig equations

1. ## Help with solving three trig equations

Hey guys, I'm in a college trig class and these questions were on my last test. I thought I had trig identities down but I barely know where to start on these ones. Can you help me out?

1. 12sin(2t)cos(3t)+20sin(2t)-cos(3t)-5=0
2. 6sec(2t)^2-13sec(2t)=6
3. 2sin(x)cos(-x)=2sin(-x)sin(x)

2. ## Re: Help with solving three trig equations

Originally Posted by Aghori2402
Hey guys, I'm in a college trig class and these questions were on my last test. I thought I had trig identities down but I barely know where to start on these ones. Can you help me out?

1. 12sin(2t)cos(3t)+20sin(2t)-cos(3t)-5=0
2. 6sec(2t)^2-13sec(2t)=6
3. 2sin(x)cos(-x)=2sin(-x)sin(x)
For the second, write \displaystyle \begin{align*} x = \sec{2t} \end{align*}, which gives \displaystyle \begin{align*} 6x^2 - 13x = 6 \end{align*}. Solve this quadratic equation for \displaystyle \begin{align*} x \end{align*}, which tells you what \displaystyle \begin{align*} \sec{2t} \end{align*} equals, which you can use to find \displaystyle \begin{align*} t \end{align*}.

For the third

\displaystyle \begin{align*} 2\sin{(x)}\cos{(-x)} &= 2\sin{(-x)}\sin{(x)} \\ 2\sin{(x)}\cos{(-x)} - 2\sin{(-x)}\sin{(x)} &= 0 \\ 2\sin{(x)}[\cos{(-x)} - \sin{(-x)}] &= 0 \\ \sin{(x)} = 0 \textrm{ or } \cos{(-x)} - \sin{(-x)} &= 0 \end{align*}

Solve each of these equations for \displaystyle \begin{align*} x \end{align*}.

3. ## Re: Help with solving three trig equations

For no. 3 Use cos(-x)=cosx and sin(-x)=-sinx
So question becomes 2sinxcosx=-2sinxsinx Cancel 2 and get sinxcosx+sinxsinx=0
sinx(cosx+sinx)=0 So sinx=0 or cosx+sinx=0 giving sinx=-cosx giving sinx/cosx=-1 That is tanx=-1
So sinx-0 or tanx=-1 Give the answers for x in whatever range you were asked.

4. ## Re: Help with solving three trig equations

Originally Posted by Aghori2402

1. 12sin(2t)cos(3t)+20sin(2t)-cos(3t)-5=0
If the question had been $12\sin(2t)\cos(3t)+20\sin(2t)-3cos(3t)-5=0$ I would have said factorise.